
\section{note of 2009.09.22}
\subsection{Equation of Three Species}
With the simplification discussed in previous, the equation of motion of order parameters \eqref{eq:zhang536} for three species can be written as (1 is the common species):
for short-range,
\begin{align}\label{eq:short}
i\hbar\pdiff{F_{12}}{t}&=\br{-\hm\nabla_{12}^2-\mu_{12}+E_{12}}F_{12}+U\;F_{12}+Y\;F_{13}\notag\\*
i\hbar\pdiff{F_{13}}{t}&=\br{-\hm\nabla_{13}^2-\mu_{13}+E_{13}}F_{13}+V\;F_{13}+Y\;F_{12}
\end{align}
and for long-range,
\begin{align}\label{eq:long}
i\hbar\pdiff{F_{12}}{t}&=\br{-\hm\nabla_{12}^2-\mu_{12}+E_{12}}F_{12}+U\;F_{12}+Y\;F_{13}+\Delta_{12}\br{G_1+G_2}+\Gamma_{23}G_1\notag\\*
i\hbar\pdiff{F_{13}}{t}&=\br{-\hm\nabla_{13}^2-\mu_{13}+E_{13}}F_{13}+V\;F_{13}+Y\;F_{12}+\Delta_{13}\br{G_1+G_3}+\Gamma_{23}G_1
\end{align}
\[U=U_{1221},\quad{} V=U_{1331},\quad{} Y=U_{1231}\]
\[\Delta_{1x}=-\br{\int{d}\vr'U_{x11x}\br{\vr-\vr'}F_{x1}(\vr-\vr')}\]
\[\Gamma_{xy}=-\br{\int{d}\vr'U_{x11y}\br{\vr-\vr'}F_{1y}(\vr-\vr')}\]

It is obvious that $F$ is not eigenstate of the hamiltonian.  However, unless something really dramatic, its short-range is very much like the two-body eigenstate.  

